Reed-Muller codes, the fourth cohornology group of a finite group, and the beta-invariant

摘要

We introduce the beta-invariant b(omega) attached to a 4-cohomology class omega is an element of H-4(G, Z), G a finite group. Roughly speaking, b(omega) keeps track of the restriction of omega to subgroups of G of order 2. If G is an elementary abelian 2-group, we observe that b defines a natural isomorphism from H4(G, Z) to the shortened third order Reed-Muller binary code. In general, restricting w to elementary abelian 2-subgroups produces an array of Reed-Muller codewords which can be exploited. We give two main applications: (a) for many of the larger sporadic simple groups, the 2-part of H4(G, Z) lies in the nilpotent radical of the cohomology ring (Proposition 4.1); (b) up to gauge equivalence, the twisted quantum double D-omega(G) has a trivial P-invariant (in the sense of quasi-Hopf algebras) if, and only if, omega is a nilpotent element in the cohomology ring (Proposition 5.2). (C) 2006 Elsevier Inc. All rights reserved.